Theory of the Nernst effect near quantum phase transitions in condensed matter and in dyonic black holes

TLDR: In this article, a general hydrodynamic theory of transport in the vicinity of superfluid-insulator transitions in two spatial dimensions described by ''Lorentz''-invariant quantum critical points was presented.

Abstract: We present a general hydrodynamic theory of transport in the vicinity of superfluid-insulator transitions in two spatial dimensions described by ``Lorentz''-invariant quantum critical points. We allow for a weak impurity scattering rate, a magnetic field $B$, and a deviation in the density $\ensuremath{\rho}$ from that of the insulator. We show that the frequency-dependent thermal and electric linear response functions, including the Nernst coefficient, are fully determined by a single transport coefficient (a universal electrical conductivity), the impurity scattering rate, and a few thermodynamic state variables. With reasonable estimates for the parameters, our results predict a magnetic field and temperature dependence of the Nernst signal which resembles measurements in the cuprates, including the overall magnitude. Our theory predicts a ``hydrodynamic cyclotron mode'' which could be observable in ultrapure samples. We also present exact results for the zero frequency transport coefficients of a supersymmetric conformal field theory (CFT), which is solvable by the anti--de Sitter (AdS)/CFT correspondence. This correspondence maps the $\ensuremath{\rho}$ and $B$ perturbations of the $2+1$ dimensional CFT to electric and magnetic charges of a black hole in the $3+1$ dimensional anti--de Sitter space. These exact results are found to be in full agreement with the general predictions of our hydrodynamic analysis in the appropriate limiting regime. The mapping of the hydrodynamic and AdS/CFT results under particle-vortex duality is also described.

Figures

FIG. 2: Nonzero temperature (T ) phase diagram at B = 0 along three vertical cuts (i.e. fixed g)

FIG. 4: Contour plot (linear scale) of the Nernst signal eN = ϑyx (Eq. 2.6) close to a quantum critical point, as a function of temperature T and magnetic field B. The parameters are the same as for Fig. 3. The signal strength in the plot ranges up to 10µV/K.

FIG. 1: Zero temperature (T = 0), zero field (B = 0) phase diagram in the vicinity of the quantum critical point described by the CFT, represented by the filled circle. The coupling g represents a parameter which tunes between a superfluid and a Mott insulator which is at a density commensurate with the underlying lattice. The chemical potential µ introduces variations in the density and ρ is difference in the density of pairs of holes in the superfluid from that in the Mott insulator. The thin dashed lines are contours of constant ρ. In the application to the cuprate superconductors, the Mott insulator with ρ = 0 could be, e.g., an insulating state at hole density δI = 1/8 in a generalized phase diagram; then ρ = (δ − δI)/(2a2), where a is the lattice spacing. The thick dotted line represents a possible trajectory of a particular compound as its hole density is decreased; note that the ground state is always a superconductor along this trajectory, even at δ = 1/8 (although there will be a dip in Tc near δ = 1/8 as is also clear from Fig. 2). Note that the parent Mott insulator with zero hole density is not shown above. This paper will describe electrical and thermal transport in the above phase diagram perturbed by an applied magnetic field B and a small density of impurities.

FIG. 3: Contour plot (with logarithmic spacing) of the thermoelectric conductivity αxy (Eq. 2.3) as a function of temperature T and magnetic field B, for parameters ~v = 47 meV Å, δ−δI = 0.025 and τimp = 10 −12s estimated for LSCO. In the ordered low temperature regime T < Tc ≈ 30K, Eq. (2.3) will receive modifications.